Analysis of functions of a single variable via multivariable calculus?

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Discussion Overview

The discussion revolves around the relationship between single-variable calculus and multivariable calculus, particularly whether studying a rigorous multivariable/vector analysis book can fill gaps left by earlier single-variable studies. Participants explore the implications of transitioning from R^1 to R^n and the necessity of foundational concepts from single-variable calculus in understanding multivariable calculus.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that R^1 is a special case of R^n, questioning if rigorous study in multivariable calculus can address gaps from single-variable calculus.
  • Another participant notes that while some theorems may not appear in vector calculus, others might be proven by reducing to the one-dimensional case, indicating a potential dependency on single-variable results.
  • Efficiency in study is emphasized, with a participant expressing a desire to avoid revisiting calculus I and II while preparing for multivariable calculus.
  • Concerns are raised about the complexities of differentiability in R^n, highlighting that even with existing partial derivatives, a function may not be differentiable.
  • Some participants introduce advanced topics such as fiber bundles and differential forms, suggesting that these could be approached without a thorough grounding in multivariable calculus.
  • There is a proposal to rephrase fundamental theorems in terms of more advanced mathematical structures, indicating a rich interplay between different areas of mathematics.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of single-variable calculus for understanding multivariable calculus, with some arguing for its importance while others suggest that advanced topics could be approached directly. The discussion remains unresolved regarding the best path for bridging these areas of study.

Contextual Notes

Participants acknowledge that certain foundational theorems from single-variable calculus, such as Rolle's theorem and the Intermediate Value Theorem, are significant and not covered in multivariable calculus. There is also mention of the complexity involved in differentiability in higher dimensions, which may not be fully addressed by single-variable concepts.

Who May Find This Useful

This discussion may be of interest to students transitioning from single-variable to multivariable calculus, educators exploring curriculum design, and those interested in advanced mathematical concepts such as fiber bundles and differential forms.

imwhatim
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Took calculus of a single variable almost a decade ago where every theorem had to be accepted without proof. Can I fill these gaps by studying a rigorous multivariable/vector analysis book? My justification for this is that R^1 is just a special case of R^n. Or am I looking at this the wrong way and proving things in R^n requires results from R^1?
 
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You are basically right, but I think there are theorems that do not occur in vector calculus. Others may be proven by a reduction to the one-dimensional case which won't be repeated. Others might use the one-dimensional case as an induction basis. Real analysis isn't properly included in vector calculus.

In the end, it depends on your goals. E.g. you could consider reading a calculus textbook as warm-up for vector calculus. If it is a matter of sources, then you could look for lecture notes on the internet: google "Calculus I + pdf".
 
fresh_42 said:
If it is a matter of sources, then you could look for lecture notes on the internet: google "Calculus I + pdf".
It's mostly a matter of being efficient with time, I didn't want to go over cal 1/2 again. I'm making a return to school and the curriculum suggests multivariable calculus. If I pick a rigorous book to go along with the course, I figured I could kill two birds with one stone.
 
imwhatim said:
It's mostly a matter of being efficient with time, I didn't want to go over cal 1/2 again. I'm making a return to school and the curriculum suggests multivariable calculus. If I pick a rigorous book to go along with the course, I figured I could kill two birds with one stone.
There is a reason that Calculus I exists. If it were covered by multivariate calculus, then it wouldn't be taught at universities. Rolle and IVT are important theorems.
 
Still, issues like differentiability in ##\mathbb R^n## are more " twisted" than in ##\mathbb R## itself, as limits must agree along all possible directions. Even if all partials exist for your function with n arguments, the function may not be differentiable.
 
WWGD said:
Still, issues like differentiability in ##\mathbb R^n## are more " twisted" than in ##\mathbb R## itself, as limits must agree along all possible directions. Even if all partials exist for your function with n arguments, the function may not be differentiable.
Once again an argument for my favored version of the differentiability definition: Weierstraß! One formula fits all.
 
There is even another level of this argument! Forget about multivariate calculus. Let's talk about fiber bundles instead!
 
Check out one of the more theoretical calculus books
Courant
Spivak
Apostal
It is a different experience, so it will not be as repetitive as you think.
 
fresh_42 said:
There is even another level of this argument! Forget about multivariate calculus. Let's talk about fiber bundles instead!
How does that fit in?
 
  • #10
WWGD said:
How does that fit in?
If you drop single-variate calculus for multivariate calculus, you could as well start with fiber bundles, sections, and differential forms to spare multivariate calculus. There is always a generalization.

That would be a nice thread / insight / or just fun: take rudimentary theorems like Rolle or IVT and rephrase them in terms of tangent bundles, sections, and differential forms, i.e. Graßman algebras. I mean they did the same thing with the fundamental theorem of calculus. It appears in so many different ways, and every version has a name: Stokes, Divergence, Gauß-Bonnet, Cauchy, and probably some more. All are about the question of how much information in which situation is already provided on the boundaries. I've read these days that even the seven bridges of Königsberg can be seen from the perspective of FTC as an early version. The association chain was: Königsberg > Euler characteristic > Gauß-Bonnet > triangularizations > Stokes > FTC.
 
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I see, so maps from ##\mathbb R^n \rightarrow \mathbb R^m## i.e., Vector Fields, viewed as sections of bundles, boundary conditions related to cconditions in the interior, etc?
 
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